# Find the location of the aerodynamic center.

An airfoil has a lift coefficient and moment coefficient of \(-0.35\) and \(-0.050\) about quarter-chord at \(-6^{\circ}\) angle of attack. The airfoil has lift coefficient and moment coefficient of \(0.70\) and \(-0.040\) at an angle of attack of \(4^{\circ}\) about the same quarter chord. Find the location of the aerodynamic center.

Aerodynamic center is a point on the airfoil for which the aerodynamically generated moment is independent of the angle of attack or the coefficient of lift. Aerodynamic center is at \(\frac{1}{4}\) chord from the leading edge for most of the low speed airfoils. For supersonic airfoils aerodynamic center lies nearer at \(\frac{1}{2}\) chord length. Location of the aerodynamic center as a fraction of the chord length is given as \[x_{ac}=-\frac{m_{0}}{a_{0}}+0.25\]\(m_{0}\) = Slope of the moment coefficient curve, \[m_{0}=\frac{-0.040-\left ( -0.050 \right )}{4^{\circ}-\left ( -6^{\circ} \right )}=0.001\textrm{ per degree}\] \(a_{0}\) = Slope of the lift coefficient curve, \[a_{0} = \frac{0.70-\left ( -0.35 \right )}{4^{\circ}-\left ( 6^{\circ} \right )} = 0.105 \textrm{ per degree}\]Therefore, aerodynamic center lies at,\[x_{ac} = -\frac{m_{0}}{a_{0}} + 0.25\]\[\Rightarrow x_{ac} = -\frac{0.001}{0.105} + 0.25 = 0.240 \textrm{ chord length}\]